
where \(M \lesssim \Lambda \)  is the scale associated with these UV-induced new operators and \(n\ge 2\) . Of particular relevance for our context is the (representative and generic) higher-derivative nature of these interactions. Including these interactions modifies the dispersion relation (REF ) and one now finds [1]}
\( \omega ^2 \sim c_{\rm GW}^2 (0) k^2 +\sum _{n\ge 2}\frac{c_n \Lambda ^6}{3M^{2n+4}}(c^2 k^2-\omega ^2)^{n-1}(\omega ^2 + {\cal O}(k^2)),\) 
